In current daily life, batteries are widely used to provide energy to consumer products, especially portable products. Among them, rechargeable batteries, including NiCd, NiMh, Li-ion and lead-acid batteries, are popular because of their advantage of rechargeability. However, these rechargeable batteries still have several disadvantages. First, recharging cycles are limited. For example, NiCd batteries normally have 1000 recharging cycles with memory effect, NiMH batteries have 500 recharging cycles and a high self-discharge rate, Li-ion batteries have 300˜500 recharging cycles. Second, the needed charging time is rather long. Normally one to several hours is needed to fully charge a battery. This limits mobility, especially for portable devices, due to the need of a connection between the battery and a fixed energy source.
Another kind of energy storage, i.e. a super capacitor, has been developed which has remarkable advantages, especially the approximately unlimited number of recharging cycles and the instant charging capability. For example, a super capacitor has more than 500,000 recharging cycles, and for each charging only a few seconds to a few minutes is needed.
However, super capacitors suffer a rapid voltage drop upon disconnecting them from a charger. A super capacitor normally is a parallel connection of a plurality of capacitors. FIG. 1 illustrates an exemplary charge model of a super capacitor, wherein, C0, C1, . . . , and Cn are capacitors, R0, R1, . . . , Rn, and RP are major series resistances of the super capacitor, Rs is the contact resistance between the charger and the super capacitor, and V is the measured voltage across the super capacitor by a voltmeter. The voltage across the super capacitor can be calculated as:Vs-c=V−i*Rs  (1)
And the voltage across an internal capacitor, for example, C0, can be calculated as:Vc0=V−i*Rs−i1*R0  (2)
From equations (1) and (2), it is easy to see that the real voltages across the super capacitor and internal capacitors should be Vs-c, Vc0, etc. And the actual stored energy
  (      E    =                  1        2            ⁢      C      *              V        2              )is determined by the capacitance of C0, C1, . . . , Cn and the corresponding voltages VC0, VC1, . . . , VCn.
Normally the super capacitor can be charged with either constant current or constant power. The charging stops upon the measured voltage V reaching a rated voltage Vrated. FIG. 2 depicts the voltage drop phenomenon. First, due to the existence of contact resistance Rs during charging, a first voltage drop Vf occurs when charging stops. Second, due to the diffusion process, such as the equilibration of excess ionic concentration, a second rapid voltage drop Vd occurs in a short period after the charging stops. For a better understanding, the diffusion process is described below. When a super capacitor is being charged, at least two processes occur near the surface of the carbon. The first process is an increase in the number of ions forming the double-layer at the carbon surface. The second process is a local increase in ionic concentration, at or near the carbon surface, which is not directly related to the double-layer. When the super capacitor disconnects from the charging circuit and undergoes self-discharging, then, apart from charge leakage due to imperfections and impurities in the double-layer, the double-layer charge will stay in place but the excess ionic concentration near the carbon surface will diffuse to an equilibrium state. Some of the ions will diffuse into the electrolyte and others will diffuse to the carbon surface where they reduce the number of excess charges in the carbon and, hence, decrease the open-circuit voltage of the capacitor.
Due to the existence of Vf and Vd, the actual voltage across the super capacitor and its internal capacitors cannot reach the rated voltage Vrated, which further contributes to the actual stored energy being smaller than the rated energy
      E    rated    =            1      2        ⁢    C    *          V      2      of the super capacitor. If the overall voltage drops about 10%, about 19% less energy is stored than rated.
Thus, there is a need to improve the charging efficiency for charging super capacitors.